For the following variables, determine whether \( a \) is a function of \( b, b \) is a function of \( a \), or neither. a is the radius of any U.S. coin and \( b \) is its denomination.

To determine the relationship between the variables \( a \) (the radius of any U.S. coin) and \( b \) (its denomination), we need to analyze how these two variables interact. 1. **Is \( a \) a function of \( b \)?** - For \( a \) to be a function of \( b \), each denomination \( b \) must correspond to exactly one radius \( a \). - In the U.S., each denomination of coin (e.g., penny, nickel, dime, quarter) has a specific radius. Therefore, for each denomination, there is a unique radius associated with it. - Thus, we can say that \( a \) is a function of \( b \). 2. **Is \( b \) a function of \( a \)?** - For \( b \) to be a function of \( a \), each radius \( a \) must correspond to exactly one denomination \( b \). - However, multiple denominations can have the same radius. For example, both the nickel and the dime have a radius of 1.35 cm. This means that knowing the radius does not uniquely determine the denomination. - Therefore, \( b \) is not a function of \( a \). In conclusion: - \( a \) is a function of \( b \). - \( b \) is not a function of \( a \).